Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order

Nonlinear boundary value problems of fractional differential equations have received a considerable attention in the last few decades. One can easily find a variety of results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional equations in the literature on the topic.The interest in the subject has beenmainly due to the extensive applications of fractional calculus in the mathematicalmodeling of several real-world phenomena occurring in physical and technical sciences; see, for example, [1–4]. An important feature of a fractional order differential operator, distinguishing it from an integer-order differential operator, is that it is nonlocal in nature. It means that the future state of a dynamical system or process based on a fractional operator depends on its current state as well as its past states. Thus, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes. This feature has fascinated many researchers, and they have shifted their focus to fractional order models from the classical integerorder models. For some recent work on the topic, we refer, for instance, to [5–9]. Recently, in [10], the authors studied sequential fractional differential equations with three-point boundary conditions. In this paper, we consider a nonlinear Dirichlet boundary value problem of sequential fractional integrodifferential equations given by


Introduction
Nonlinear boundary value problems of fractional differential equations have received a considerable attention in the last few decades.One can easily find a variety of results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional equations in the literature on the topic.The interest in the subject has been mainly due to the extensive applications of fractional calculus in the mathematical modeling of several real-world phenomena occurring in physical and technical sciences; see, for example, [1][2][3][4].An important feature of a fractional order differential operator, distinguishing it from an integer-order differential operator, is that it is nonlocal in nature.It means that the future state of a dynamical system or process based on a fractional operator depends on its current state as well as its past states.Thus, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes.This feature has fascinated many researchers, and they have shifted their focus to fractional order models from the classical integerorder models.For some recent work on the topic, we refer, for instance, to [5][6][7][8][9].Recently, in [10], the authors studied sequential fractional differential equations with three-point boundary conditions.

Linear Fractional Differential Equations
For  ∈ (1, 2], we consider the following linear fractional differential equation: where    denotes the Caputo fractional derivative of order .Rewriting (1) as    (() +    −1 ()) = ℎ(), we can write its solution as where  0 ,  1 are arbitrary constants.Now, (6) can be expressed as Differentiating (7), we obtain which can alternatively be written as Integrating from 0 to , we have where  and  are arbitrary constants, and Lemma 1.The unique solution of the linear equation (5) subject to the Dirichlet boundary conditions (2) is given by Proof.Observe that the general solution of ( 5) is given by (10).Using the given boundary conditions in (10), we find that Substituting the values of  and  in (10) yields the solution (12).This completes the proof.
In the next two lemmas, we present the unique solutions of (5) with different kinds of boundary conditions.We do not provide the proofs for these lemmas as they are similar to that of Lemma 1.

Lemma 2. The unique solution of the problem (5)-(3) is given by
Lemma 3. The unique solution of (5) with the boundary conditions (4) is (15)
In view of Lemma 1, we transform problem (1)-( 2) to an equivalent fixed point problem as where V : P → P is defined by ×  (,  ()) ) . ( In a similar manner, we can define a fixed point operator V 1 : P → P for the nonlinear problem (1)-(3) as follows: A fixed point operator V 2 : P → P for the nonlinear problem ( 1)-( 4) is defined by We only present the existence results for the problem (1)- (2).Observe that problem (1)-( 2) has solutions if the operator equation ( 16) has fixed points.
Proof.Let us define  = max{ 1 ,  2 }, where  1 ,  2 are finite numbers given by sup By the given assumption,  < 1/, V is a contraction.Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).
Our next existence result relies on Krasnoselskii's fixed point theorem.
For , V ∈   , we find that which is independent of  and tends to zero as  2 →  1 .Thus,  1 is relatively compact on   .Hence, by the Arzelá-Ascoli theorem,  1 is compact on   .Thus, all the assumptions of Lemma 5 are satisfied.So, by the conclusion of Lemma 5, problem (1)-( 2) has at least one solution on [0, 1].Now, we show the existence of solutions for the problem (1)-( 2) via Leray-Schauder alternative.
Lemma 7 (nonlinear alternative for single valued maps, see [12]).Let  be a Banach space,  a closed, convex subset of ,  an open subset of , and 0 ∈ .Suppose that  :  →  is a continuous, compact (that is, () is a relatively compact subset of ) map.Then, either (i)  has a fixed point in , or (ii) there is a  ∈  (the boundary of  in ) and  ∈ (0, 1) with  = ().
We show that V maps bounded sets into bounded sets in ([0, 1], R).For a positive number , let   = { ∈ ([0, 1], R) : ‖‖ ≤ } be a bounded set in ([0, 1], R).Then, Obviously, the right hand side of the previous inequality tends to zero independently of  ∈   as  2 −  1 → 0. As V satisfies the previous assumptions, therefore it follows by the Arzelá-Ascoli theorem that V : ([0, 1], R) → ([0, 1], R) is completely continuous.The proof will be complete by the application of the Leray-Schauder nonlinear alternative (Lemma 7) once we establish the boundedness of the set of all solutions to equations  = V for  ∈ (0, 1).
Let  be a solution.Then, for  ∈ [0, 1], and using the computations in proving that V is bounded, we have Note that the operator V :  → ([0, 1], R) is continuous and completely continuous.From the choice of , there is no  ∈  such that  = V() for some  ∈ (0, 1).Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 7), we deduce that V has a fixed point  ∈  which is a solution of the problem (1)-( 2).This completes the proof.